Can Arithmetic Progressions Form Infinite Relatively Prime Subsequences?

[robin_vanp]

a question came up

"show that the arithmetic progression ax+b contains an infinite subsequence (not necessarily a progression), every two of whose elements are relatively prime."

i have a hunch that the chinese remainder theorem has something to do with this, but I'm not sure how. any thoughts?

 

[DeadWolfe]

Is that true? What if a=2, b=o?

 

[robin_vanp]

sorry, assuming a, b are non zero

 

[DeadWolfe]

Then a=2, b=2 is a counterexample. I think you really need that a and b are coprime, in which case the sequence actually contains infinitely many primes.

 

[robin_vanp]

right again. its actually a two part question so it says on the top that (a,b) = 1, i forget to mention; if so (now that we finally got the problem) how is the CRT applicable here?

 

[robin_vanp]

and deriving some sort of solution that does not employ dirichlet's theorem, i think, because then that would be obvious; i really do not know how the CRT can be used here.